Section P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

Objectives

• Evaluate algebraic expressions.

• Use mathematical models.

• Find the intersection and union of two sets.

• Recognize subsets of the real numbers.

• Use inequality symbols.

• Evaluate absolute value and use it to express distance.

• Identify properties of real numbers.

• Simplify algebraic expressions.

Algebraic Expressions

Definition and Evaluation

An algebraic expression is a combination of variables and numbers using the operations of addition, subtraction, multiplication, division, as well as powers or roots. A variable is a letter used to represent various numbers. Evaluating an algebraic expression means to find its value for a given value of the variable.

• Exponential Expression: An expression of the form is called an exponential expression.

Order of Operations

To evaluate algebraic expressions correctly, follow the order of operations:

1. Perform operations within the innermost parentheses first, working outward. If the expression involves a fraction, treat the numerator and denominator as if each is enclosed in parentheses.

2. Evaluate all exponential expressions.

3. Perform multiplications and divisions as they occur, working from left to right.

4. Perform additions and subtractions as they occur, working from left to right.

Example: Evaluating an Algebraic Expression

Evaluate for .

• Substitute :

Formulas and Mathematical Models

Definitions

• An equation is formed when an equal sign is placed between two algebraic expressions.

• A formula is an equation that uses variables to express a relationship between two or more quantities.

• Mathematical modeling is the process of finding formulas to describe real-world phenomena. The formulas, together with the meaning assigned to the variables, are called mathematical models.

Example: Using a Mathematical Model

The formula models the average cost of tuition and fees, , for public U.S. colleges for the school year ending years after 2000. Find the average cost for the year ending in 2016.

• Substitute into the formula:

• Thus, the average cost was $8713.

• If the actual cost was $8778, the formula underestimates by $8778 - 8713 = .

Sets and Set Notation

Definitions

• A set is a collection of objects, called elements.

• Sets are denoted with braces, e.g., {3, 4, 5, ...}.

• The empty set (or null set) has no elements and is denoted by or {}.

Set Notation

• Roster method: Lists all elements, e.g., {1, 2, 3, 4, 5}.

• Set-builder notation: Describes elements, e.g., .

Intersection and Union of Sets

• Intersection (): The set of elements common to both sets A and B.

• Union (): The set of elements that are in A, in B, or in both.

Example: Intersection and Union

• Intersection:

• Union:

Subsets of the Real Numbers

Classification of Real Numbers

• Natural Numbers:

• Whole Numbers:

• Integers:

• Rational Numbers: Numbers that can be expressed as , where and are integers and .

• Irrational Numbers: Numbers that cannot be expressed as a quotient of integers (e.g., , ).

The set of real numbers is the union of the set of rational numbers and the set of irrational numbers.

The Real Number Line

The real number line is a graphical representation of all real numbers. The point labeled 0 is called the origin. Numbers to the right of the origin are positive; numbers to the left are negative. The distance from 0 to 1 is called the unit distance.

Inequality Symbols

Inequality symbols are used to compare the sizes of real numbers:

• means "a is less than b"

• means "a is less than or equal to b"

• means "a is greater than b"

• means "a is greater than or equal to b"

The symbols always point to the lesser of the two numbers when the statement is true.

Absolute Value and Distance

Definition of Absolute Value

The absolute value of a real number , denoted , is the distance from 0 to on the number line. It is always nonnegative.

• if

• if

Example: Evaluating Absolute Value

Distance Between Points on the Real Number Line

The distance between two points and on the real number line is given by or .

Example: Distance Between Two Points

• Distance between and is

Properties of Real Numbers

Simplifying Algebraic Expressions

To simplify an algebraic expression, combine like terms (terms with exactly the same variable factors) and remove parentheses. An expression is simplified when all like terms have been combined and all parentheses have been removed.